All Externally Owned Accounts on the Ethereum blockchain are controlled by a random 32-byte private key. An account's public key and public Ethereum address are derived from this private key through Elliptic Curve operations. BIP-39 proposed to encode the 32-byte key with a mnemonic phrase or seed phrase. "private_key" is a visual representation of a private key composed of the 2048 possible words that can be in a mnemonic seed phrase as defined by BIP-39.
